The prime minister’s spokesman repeated the line on Wednesday that the government is determined to secure “legally binding changes” to the Irish backstop, despite the attorney general, Geoffrey Cox, returning empty-handed from the talks.

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Shortly before leaving Brussels, he conceded “strong views” had been expressed during three hours of “robust” discussions.

Downing Street said the talks had been “difficult”, but stressed the vote would take place on Tuesday, as committed by May. If it is lost, MPs will vote on successive days on whether to block a no-deal Brexit and whether to extend the departure date.

With it increasingly being assumed No 10 will not secure significant EU concessions on the backstop, May is expected to try to sell her plan to MPs and the public later in the week, potentially with a speech.

There are no plans as yet for the prime minister or Cox to return to Brussels, but it is understood this could happen if required. Sunday night is the final deadline for any changes, as the government needs to publish and print copies of deal documents on Monday, and publish the motion MPs will then vote on.

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The two sides are seeking to find a set of assurances on the temporary nature of the Irish backstop to win over MPs to the prime minister’s deal when it returns to the House of Commons.

Cox said: “There are very sensitive discussions. We are into the meat of the matter now. We have put forward some proposals, very reasonable proposals. We are now into the detail of the discussion.”

The government’s chief legal adviser, who had travelled with the Brexit secretary, Stephen Barclay, to Brussels, added: “I can’t reveal the discussions. These are private and confidential discussions. Both sides have exchanged robust, strong views and we are now facing the real discussions. Talks will be resuming soon.”

Cabinet ministers are pessimistic about the prospects of wooing enough waverers in parliament to win the second meaningful vote.

May will then come under intense pressure to offer MPs a free vote on a no-deal Brexit, with her cabinet deeply divided. The opportunity to vote on this was only secured after scores of frontbenchers made it clear they were willing to resign, and the cabinet ministers Amber Rudd, Greg Clark and David Gauke wrote to the Daily Mail rejecting the idea of a no-deal departure.

Senior EU diplomats said the bloc’s deputy chief negotiator, Sabine Weyand, offered a “gloomy” analysis of the talks. According to a diplomatic note, Weyand told the ambassadors: “Cox’s asks are going well beyond where Barnier can go.”

The “thorniest issue” is described as a request for the “independent review mechanism”, already in the agreement, to be outside the jurisdiction of the European court of justice.

“The assurances on independence that the QC [Cox] seeks would mean substantial carve out of EU law,” Weyand told the ambassadors. “No indication this would bring the vote home. Such guarantees let alone indications were not given.”

“Tomorrow parties will review whether new meetings are considered helpful,” the note adds. “The general feeling: UK is seeking substantial assurances without any assurance these will in any way actually work.”

The EU’s chief negotiator, Michel Barnier, also gave a downcast progress report to officials on Wednesday morning, including the European commission president, Jean-Claude Juncker.

A European commission spokesman disclosed that Barnier had “informed the commissioners that while the talks take place in a constructive atmosphere, discussions have been difficult”.

The latest negotiations have been focused on turning a series of pledges made in a letter in December by Juncker and his European council counterpart, Donald Tusk, into legally binding commitments.

Those include stating the EU’s “firm determination” to have an alternative to the backstop ready before the end of 2020, to avoid it being triggered.

Should the customs union it envisages come into force, both sides would set the “objective of making this period as short as possible”.

Such commitments could be used in arbitration by the UK should there be any doubt about the “good faith” of the EU negotiators in subsequent talks.

But the EU in turn wants to avoid any legal add-on to the withdrawal agreement that undermines the terms of the backstop, which would keep the UK in a customs union and Northern Ireland in the single market if there were no other alternative to avoiding a hard border in Ireland.

There are concerns, given the debate in Westminster and the prime minister’s insistence that the UK should be able to enjoy the benefits of frictionless trade without being in the customs union, that the EU is setting itself an unachievable goal of making such swift progress in the future trade talks.

The British side is seeking to convince the EU’s negotiators that this week offers the best chance of winning over parliament, and reversing the 230-vote defeat of May’s withdrawal agreement in January.

There are concerns among UK officials that the EU side is assuming there will a better opportunity for success once a one-off extension of article 50 is triggered, and MPs face the choice of a deal or no deal at a later vote.

The prospect of MPs being given an opportunity to force a softer Brexit were boosted by a government defeat in the House of Lords on Wednesday night on the trade bill.

The Department for Exiting the EU sought to underline its determination to press ahead on Wednesday, by announcing that it would create three new expert panels to investigate “alternative arrangements” to checks at the border between Northern Ireland and the Republic, as part of a joint UK-EU “workstream”.

One panel will be made up of “technical experts in trade and customs”, a second of business and trade unions, and a third of MPs, DExEU said.

They will be tasked with looking at international best practice, and examining ways technology can be used to make border checks less onerous.